Abstract

In this article, we analyze representations for the product Ym1l1(∇)Fm2l2(r) with Yml(∇) specifying a solid harmonic whose argument is the nabla operator ∂/∂r instead of the vector r. Since both Ym1l1(∇) and Fm2l2(r) are irreducible spherical tensors, we can use angular momentumalgebra for evaluating the product. Accordingly, the problem of finding a representation for the product is reduced to the determination of the radial functions generated by the product. Analytical expressions for these radial functions are derived by direct differentiation and with the help of Fourier transforms. Closely related to the spherical tensor gradient Yml(∇) is the spherical delta function δml(r). We derive new representations for δml by considering convolution integrals involving B functions. These functions are closely related to the modified Bessel functions and also to the Yukawa potential e−αr/r. We show that the definition of the B functions can be extended to include a large class of derivatives of the delta functions, where the spherical delta function is just a special case.